Antigenic distance and cross-immunity, invasibility and coexistence of pathogen strains in an epidemiological model with discrete antigenic space.

In models of pathogen interaction and evolution discrete genotypes in the form of bit strings may be mapped to points in a discrete phenotype space based on similarity in antigenic structure. Cross-immunity between strains, that is the reduction in susceptibility to strain A conferred to a host by infection with strain B, can then be defined for pairs of points in the antigenic space by a specified function. Analysis of an SIR type model shows that, if two strains are at equilibrium, the shape of the cross-immunity function has a strong influence on the invasion and coexistence of a third strain and, consequently, the expected evolutionary pathway. A function that is constant except for discontinuities at the end points is expected to result in the accumulation of diversity until a pair of discordant strains occurs that can, depending on parameter values, exclude all other strains. For a function of the form f(h)=hq, where h is the antigenic distance between two strains, invasion and coexistence is always possible if q≤1 and little antigenic structure is expected in the pathogen population. However, if q>1 invasion and coexistence may be impossible, depending on parameter values, and the pathogen population is expected to show significant antigenic structuring. In addition to illuminating the role of cross-immunity in pathogen evolution, this analysis indicates that the choice of cross-immunity function, the representation of immunity acquired from multiple previous infections and the number of elements used to characterize the antigenic space must be carefully considered in the development and interpretation of more sophisticated models of pathogen dynamics and evolution.